The present invention relates to imaging systems and methods, and, more particularly, to a system and method for imaging a remote target through a turbulent medium. An exemplary application is ground-based imaging of satellites through atmospheric turbulence.
Satellite imaging can be effected by illuminating a target satellite with a laser and detecting the reflections. Where the laser, the detector, or most commonly both, are ground based, the resolution of a satellite image is limited by the effects of atmospheric turbulence.
"Atmospheric turbulence" refers to the random time-varying non-uniformities in the refractive index of the atmosphere. The effects of atmospheric turbulence are commonly experienced as the twinkling of stars. The non-uniformities in the atmosphere's refractive index distort an otherwise disk-shaped star image so that the star appears to have points. The time-variations in these non-uniformities cause the points to change.
Where the illumination source and the detectors are both based on the ground, atmospheric turbulence distorts both the illumination beam and the reflection so that distortion is compounded. The basic effect is the same: the desired image is "smeared" relative to an image that would be obtained in the absence of turbulence.
Adaptive optics systems have been developed which work with reflected sunlight from the target to be imaged. They introduce a wavefront distortion to the light from the satellite, which light has been subject to turbulence induced distortion, and then form an image of the satellite using this light. The adaptive optics systems are designed so as to be able to sense the instantaneous nature of the turbulence induced wavefront distortion and adjust the distortion they introduce so that it exactly cancels the distortion due to turbulence. This "predetection" compensation of the wavefront before image formation and detection allows nearly diffraction-limited images to be produced.
Adaptive optics systems can be bulky and expensive. Generally, they include a wavefront distortion sensor, a large amount of high speed electronics, and a deformable mirror, which is deformed by an array of piezoelectric crystals according to computations based on sensor readings. The bulk and expense of deformable mirrors and piezoelectric arrays, as well as of the wavefront sensor and high speed electronics are avoided in approaches which use statistical post-detection processing rather than predetection processing to obtain the desired information about a satellite.
Laser correlography is one such post-processing technique which provides the power spectral density of an image using the statistics of laser speckle. "Laser speckle" describes the distribution of the laser reflection from the inherently microscopically rough surfaces of the target as measured over a very brief detection interval. Over a sufficiently brief duration, the detected distribution is characterized by a random distribution of discrete spots or "speckles". This random distribution changes over successive time periods due to very slight amounts of target orientation changes to yield a uniform distribution over a longer exposure.
While the individual laser speckle distributions resemble the target image's power spectral density no more than the smear obtained by adding the distributions, they do contain considerable high spatial frequency content which is obscured over longer exposures. By detecting and storing the individual laser speckle distributions, the successive distributions can be combined statistically without losing such information.
More particularly, the statistics of all of the individual laser speckle patterns exclude spatial frequencies which would not be represented in the ideal image of a target with finite extent. Likewise, if certain spatial frequencies are stronger in the ideal image of the target, then they will be correspondingly stronger in the statistics of the laser speckle patterns.
Laser correlography exploits these facts to determine the amplitude of all spatial frequency components of the ideal image. The power spectral density associated with a given spatial frequency of the image can be determined from the covariance of the intensities falling on a pair of detectors, the spacing of which detectors corresponds to that spatial frequency. The statistical data for the covariances can be obtained by multiple exposures and/or multiple pairs of detectors with the same separation.
Using a sufficient array of detectors, one can obtain the ideal image's complete power spectral distribution, i.e., the average of the square of the amplitude of each spatial frequency component of the ideal image. The inverse Fourier transform of the power spectral density yields the autocovariance of the ideal image.
An autocovariance of the target object is insufficient to produce an image of the object. The square root of the power spectral distribution can be used to determine an amplitude associated with each spatial frequency. However, the relative phases of the spatial frequency components are not obtained. In constructing an image, these relative phases are considered at least as important as the intensity information.
The problem with the failure of laser correlography to provide phase information has its parallel in "white light" speckle imaging systems, for example star imaging systems. (Herein, light is "white" if during an exposure time the oscillations at different optical frequencies have gone through different numbers of complete cycles.) Short exposures at the focal plane of a telescope yield star speckle patterns. The power spectral density of a star's image can be determined using white light speckle statistics, as demonstrated by A. Labeyrie in "Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in stars images", Astron. Astrophsys. 6, 85-87 (1970). To obtain the power spectral distribution in a white light speckle imaging system, the intensities of a white light speckle distribution at each point of a focal plane are detected. The resulting distribution is Fourier transformed to yield a spatial frequency distribution. The amplitudes of the spatial frequency distribution are squared. Since the intensities are represented by complex numbers, squaring involves multiplying an intensity by its complex conjugate i.e. C.times.C*. Successive squared spatial frequency distributions are averaged and normalized to yield a normalized autocorrelation function of the image. The inverse Fourier transform of the autocorrelation function yields the power spectral density for the target image. Again, the phase information required to obtain the image is absent.
Subsequently, it was suggested that phase information could be inferred from a power spectral density. An approach developed by J. Fienup basically uses a process of elimination in excluding all phase distributions which would imply a negative intensity somewhere in the image, given the statistically obtained power spectral distribution. Ideally, this approach yields a phase distribution which is unique, except that it is indistinguishable from its mirror image. This phase distribution can be combined with the intensity information to yield an image.
Fienup's approach has been considered for application to the power spectral density obtained for laser imaging systems as well. While the approach is not well understood, it does sometimes allow images to be constructed in both white light and laser imaging systems. However, it is not generally satisfactory, and seems particularly sensitive to noise.
A superior methodology using the "Knox-Thompson algorithm" has been developed for white light speckle imaging systems. See J. C. Fontanella and A. S/e/ ve, Journal of the Optical Society of America, Vol. 4, No. 3, pp. 438-448 (1987). Labeyrie's method is followed to obtain an intensity distribution of spatial frequencies. A similar process is used to obtain the phases. However, instead of squaring spatial frequency amplitudes, nearby spatial frequency amplitudes are multiplied. The operation can be expressed as C.times.C'*, where C and C' are the amplitudes associated with two unequal but close spatial frequencies. Averaging and normalizing the resulting products of spatial frequency amplitudes yields a distribution of phase differences between the spatial frequencies components of the ideal image. For example, the phase difference between C and C' is obtained.
An absolute phase distribution can then be determined by summing phase differences. For example, the absolute phase, .phi., of a 5 cycles per meter spatial frequency in an arbitrary orientation can be obtained by summing the phase differences between 5 and 4 cycles per meter, 4 and 3 cycles per meter, 3 and 2 cycles per meter, 2 and 1 cycles per meter, and 1 and 0 cycles per meter. The spatial frequency distribution of the target image is then obtained in the form of Ae.sup.i.phi. for each spatial frequency, where A is a real number determined using LaBeyrie's technique, and .phi. is the phase determined by offset multiplication and summing of differences, i.e., by the Knox-Thompson algorithm. This spatial frequency distribution can be inverse Fourier transformed to obtain the desired image.
The desirability of finding a Knox-Thompson analog for laser speckle imaging has been recognized. However, other than the fact that they both start with speckle distributions and yield power spectral densities, laser and white light speckle imaging methods are quite distinct. There is no step in laser correlography corresponding to the squaring of spatial frequency distributions in Labeyrie's technique. Thus, there is no step to replace the squaring process with the offset multiplication process as in the Knox-Thompson method.
The significance of this non-correspondence is that a satisfactory system and method for imaging unknown satellites using laser speckle measurements has remained elusive. What is needed is a method for obtaining from the laser speckle data the phase information which can be used independently or in combination with the intensity information such as that provided by laser correlography to construct an image of a target despite atmospheric turbulence.